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SLIDE SHOW INSTRUCTIONS This presentation is completely under your control.

SLIDE SHOW INSTRUCTIONS This presentation is completely under your control. This lesson will show only one step at a time, to see the next step you must press a key. (Actual names written on a key are in green ) TO STOP THE SLIDE SHOW : press ‘escape’ ( Esc , top left of keyboard)

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SLIDE SHOW INSTRUCTIONS This presentation is completely under your control.

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  1. SLIDE SHOW INSTRUCTIONS • This presentation is completely under your control. • This lesson will show only one step at a time, • to see the next step you must press a key. • (Actual names written on a key are in green) • TO STOPTHE SLIDE SHOW: press ‘escape’ (Esc, top left of keyboard) • TO MOVE FORWARD: press the “spacebar” or Enter • (PageDn,  , , also work) • TO MOVE BACKWARD: press the  key • (PageUp, or also work)

  2. Polynomial Long Division

  3. Step 2: Look at the first term on the outside and the inside Polynomial Long Division Divide : 2x3 + 5x2 - x + 6 by x + 3 Step 1: Write the problem using a division symbol

  4. ? Put 2x2 on the top Step 3:The outside term (x) was multiplied by (something) to equal (2x3), the inside term. We must figure out what that (something) was. x times (what?) = 2x3 2x2 Well, we started with one x and we ended up with x3, so we picked up two more x’s or x2. Also, we now have a 2 that we didn’t have before. So, the term we are looking for is 2x2

  5. Be sure to change the signs of every term. - 2x3 + 6x2 The next step is subtraction so we have: -(2x3 + 6x2) = -2x3- 6x2 2x2 Multiply the term you just wrote on top by the outside terms. 2x2(x + 3) = 2x3 + 6x2 (This answer will be written in the next line, under the correct powers)

  6. Subtract (The first terms should always cancel out) -2x3 - 6x2 Bring down the next term 2x2 - x - x - x2 Step 2: what did we multiply the outside term by to get the inside term. Step 3: Write this term on top Now we will repeat the whole process again. Step 1: look at the first terms

  7. Subtract (The first terms should always cancel out) Be sure to change the signs of every term. -2x3 - 6x2 Bring down the next term + x2 + 3x 2x2 - x - x - x2 + 2x + 6 Step 4: Multiply this new term by the outside terms Step 5: Change the signs & write the answer under the current inside term

  8. Subtract (The first terms should always cancel out) Be sure to change the signs of every term. -2x3 - 6x2 + x2 + 3x - 2x - 6 ANSWER IS ON TOP Repeat Steps 1-5 Step 1: Look at first terms Step 2: What did we multiply by? Step 3: Write this above the line Step 4: Multiply new term by outside terms Step 5: Change signs & subtract 2x2 - x + 2 - x - x2 + 2x + 6 0

  9. PROBLEM: Terms out of descending order SOLUTION: Rearrange terms into descending order Polynomial Long Division Divide : 3x5 - 17x4 - 15x3 + 4x + 54x2 - 24 by x - 6

  10. Step 2: Look at the first term on the outside and the inside Polynomial Long Division Divide : 3x5 - 17x4 - 15x3 + 54x2 + 4x - 24 by x - 6 Step 1: Write the problem using a division symbol

  11. ? Put 3x4 on the top Step 3:The outside term (x) was multiplied by (something) to equal (3x5), the inside term. We must figure out what that (something) was. x times (what?) = 3x5 3x4 Well, we started with one x and we ended up with x5, so we picked up four more x’s or x4. Also, we now have a 3 that we didn’t have before. So, the term we are looking for is 3x4

  12. Multiply the term you just wrote on top by the outside terms. 3x4(x - 6) = 3x5 - 18x4 3x5 - 18x4 - + Be sure to change the signs of every term. 3x4 The next step is subtraction so we have: -(3x5 - 18x4) = -3x5+ 18x4

  13. Subtract (The first terms should always cancel out) -3x5+ 18x4 Bring down the next term 3x4 + x3 - 15x3 + x4 Step 2: what did we multiply the outside term by to get the inside term. Step 3: Write this term on top Now we will repeat the whole process again. Step 1: look at the first terms

  14. Subtract (The first terms should always cancel out) Be sure to change the signs of every term. -3x5+ 18x4 Bring down the next term - x4 + 6x3 Step 4: Multiply this new term by the outside terms Step 5: Change the signs & write the answer under the current inside term 3x4 + x3 - 15x3 + x4 - 9x3 + 54x2

  15. Subtract (The first terms should always cancel out) -3x5+ 18x4 Bring down the next term - x4 + 6x3 - 9x3- 54x2 Be sure to change the signs of every term. Repeat Steps 1-5 Step 1: Look at first terms Step 2: What did we multiply by? Step 3: Write this above the line Step 4: Multiply new term by outside terms Step 5: Change signs & subtract 3x4 + x3 - 9x2 - 15x3 + x4 - 9x3 + 54x2 0x2 + 4x

  16. Subtract (The first terms should always cancel out) -3x5+ 18x4 Bring down the next term - x4 + 6x3 - 9x3 - 54x2 Be sure to change the signs of every term. - 0x2 + 0x Repeat Steps 1-5 Step 1: Look at first terms Step 2: What did we multiply by? Step 3: Write this above the line Step 4: Multiply new term by outside terms Step 5: Change signs & subtract 3x4 + x3 - 9x2 + 0x - 15x3 + x4 - 9x3 + 54x2 + 0x2 + 4x + 4x - 24

  17. Subtract (The first terms should always cancel out) -3x5+ 18x4 - x4 + 6x3 - 9x3 - 54x2 Be sure to change the signs of every term. - 4x+ 24 - 0x2 + 0x ANSWER IS ON TOP Repeat Steps 1-5 Step 1: Look at first terms Step 2: What did we multiply by? Step 3: Write this above the line Step 4: Multiply new term by outside terms Step 5: Change signs & subtract 3x4 + x3 - 9x2 + 0x + 4 - 15x3 + x4 - 9x3 + 54x2 + 0x2 + 4x + 4x - 24 0

  18. The division problems we just worked ended with zero remainders. Now let’s work a problem that ends with a remainder that’s not a zero. We’ll also throw in a couple of fractions so you can see how they are handled.

  19. Divide : 6x4 - 3x3 - x - 5 by 2x - 3 Polynomial Long Division Step 1: Write the problem using a division symbol This polynomial (inside) has a power missing (x2). This is a common occurrence in polynomial long division problems. Watch out for missing powers!

  20. Divide : 6x4 - 3x3 - x - 5 by 2x - 3 SOLUTION: Insert the missing power with a zero coefficient Polynomial Long Division PROBLEM: Missing the x2 term

  21. Divide : 6x4 - 3x3 - x - 5 by 2x - 3 Step 2: Look at the first term on the outside and the inside Polynomial Long Division Step 1: Write the problem using a division symbol

  22. ? Put 3x3 on the top Step 3:The outside term (x) was multiplied by (something) to equal (6x4), the inside term. We must figure out what that (something) was. x times (what?) = 6x4 3x3 Well, we started with one x and we ended up with x4, so we picked up three more x’s or x3. Also, the 2 changed into a 6, so we multiplied by 3. So, the term we are looking for is 3x3

  23. - + Be sure to change the signs of every term. 6x4 - 9x3 Multiply the term you just wrote on top by the outside terms. 3x3(2x - 3) = 6x4 - 9x3 3x3 The next step is subtraction so we have: -(6x4 - 9x3) = - 6x4+ 9x3

  24. Subtract (The first terms should always cancel out) -6x4 + 9x3 Bring down the next term 3x3 + 3x2 + 0x2 6x3 Step 2: what did we multiply the outside term by to get the inside term. Step 3: Write this term on top Now we will repeat the whole process again. Step 1: look at the first terms

  25. Subtract (The first terms should always cancel out) Be sure to change the signs of every term. -6x4 + 9x3 Bring down the next term - 6x3 + 9x2 - x Step 4: Multiply this new term by the outside terms Step 5: Change the signs & write the answer under the current inside term 17 2 3x3 + 3x2 6x3 + 0x2 + 9x2

  26. 9 + x 2 Subtract (The first terms should always cancel out) -6x4 + 9x3 Bring down the next term Be sure to change the signs of every term. - 6x3 + 9x2 - x - 9x2+ x 27 17 2 2 Repeat Steps 1-5 Step 1: Look at first terms Step 2: What did we multiply by? Step 3: Write this above the line Step 4: Multiply new term by outside terms Step 5: Change signs & subtract 3x3 + 3x 6x3 + 0x2 + 9x2 5x - 5 If the coefficient of the outside term, 2x, does not go evenly into the coefficient of the inside term, 9x2, then the number that goes on top will be: (inside/outside)= 9/2

  27. 9 5 5 + + 2 2 2x-3 Subtract (The first terms should always cancel out) DIVISOR -6x4 + 9x3 Be sure to change the sign of every term. - 6x3 + 9x2 - x - 9x2+ x - 5x + 27 15 17 2 2 2 REMAINDER The remainder is written as a fraction. the remainder over the divisor (outside polynomial) Repeat Steps 1-5 Step 1: Look at first terms Step 2: What did we multiply by? Step 3: Write this above the line Step 4: Multiply new term by outside terms Step 5: Change signs & subtract ANSWER IS ON TOP 3x3 + 3x + x 6x3 + 0x2 + 9x2 5x - 5 5 No more terms to bring down, this (5) is the remainder

  28. Answers: 1) 5x - 1 2) 6x + 5 3) 4x2 + 7x + 12 + 4) x2 - 2x - 7 5) 5x2 + 10x + 22 + Practice Problems: (Hit enter to see the answers) Divide using Polynomial Long Division 1) 15x2 + 22x - 5 by 3x + 5 2) 12x2 - 32x - 35 by 2x - 7 3) 4x3 - 2x - x2 + 6 by x - 2 4) 3x3 - 5x2 - 23x - 7 by 3x + 1 5) 5x3 + 2x - 3 by x - 2

  29. End of Tutorial Go to www.greenebox.com for more great math tutorials for your home computer Questions? send e-mail to: lgreene1@satx.rr.com

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